Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.615 - 0.788i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.396 + 0.918i)3-s + 4-s + (0.286 − 0.957i)5-s + (−0.396 − 0.918i)6-s + (−0.686 − 0.727i)7-s − 8-s + (−0.686 + 0.727i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (0.396 + 0.918i)12-s + (0.286 − 0.957i)13-s + (0.686 + 0.727i)14-s + (0.993 − 0.116i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.396 + 0.918i)3-s + 4-s + (0.286 − 0.957i)5-s + (−0.396 − 0.918i)6-s + (−0.686 − 0.727i)7-s − 8-s + (−0.686 + 0.727i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (0.396 + 0.918i)12-s + (0.286 − 0.957i)13-s + (0.686 + 0.727i)14-s + (0.993 − 0.116i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.615 - 0.788i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.615 - 0.788i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.615 - 0.788i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1323, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.615 - 0.788i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.139635048 - 0.5559836497i$
$L(\frac12,\chi)$  $\approx$  $1.139635048 - 0.5559836497i$
$L(\chi,1)$  $\approx$  0.8268710644 - 0.08518544257i
$L(1,\chi)$  $\approx$  0.8268710644 - 0.08518544257i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.6498600767875676372878422215, −18.03620091657976268270938359094, −17.49219499913721950906558651824, −16.74931288352370409834040319243, −15.75087938693492187784831659392, −15.29773747877434597599415163691, −14.47906900935767830540123600364, −13.97265075016154819262735996459, −12.95061190971215378180486398702, −12.15410468995516107062570683454, −11.8970209521566729946088686674, −10.883814130881915661780683051381, −10.0621391585176649276764766599, −9.577874851751015897372086995031, −8.85395144370726779474983628698, −8.05507992635802151667381108474, −7.41100006438154856622685938254, −6.73754413967347169203246050868, −6.20796536426837978326486122552, −5.66108282128020969901840362264, −3.987396860041971656103416820778, −3.08509898416260502763111438240, −2.32426586810049124978247155895, −2.04214177304099749712342495690, −0.89867282112762449331822278270, 0.63596122214203270190330819978, 1.132199665198101928154039393333, 2.60902047546838309184525868472, 3.17294731873825542528968929387, 3.827212775738006322215159547851, 5.14576951823843962433764268851, 5.489328272900878404584859544491, 6.44635036525860457227607838840, 7.52758529300737931265977889784, 8.0751921046824638805572992312, 8.76215538280547822569961859413, 9.41709593585335852760707880397, 9.96755175633697075483192791919, 10.460636464365365665649072522, 11.29226249845422266326806765476, 11.97156642268050642005393030765, 13.00765565744667512912928110378, 13.65275878719470217635036658603, 14.227536680375164146312487647316, 15.43530323409170862222178898417, 15.947974140852336179584672747015, 16.279572748176567155673609029582, 16.83565761255185937830034688655, 17.64584320318513314666692985513, 18.20291246053535588002069068949

Graph of the $Z$-function along the critical line